The Modulational Instability for a Generalized Korteweg–de Vries Equation

Abstract

We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength.A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evan's function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first index counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second index provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Hǎrǎguş and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator.We sketch the connection of this calculation to a study of the linearized operator -in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests a geometric interpretation attached to the vanishing of the modulational instability index previously mentioned.

On the stability of periodic solutions of the generalized Benjamin–Bona–Mahony equation

“…Most results in the periodic case fall into one of the following two categories: spectral stability under localized or bounded perturbations [6][7][8][9], and nonlinear (orbital) stability under periodic perturbations [10][11][12][13][14]. The spectral stability results rely on a detailed analysis of the spectrum of the linearized operator in a given Hilbert space representing the class of admissible perturbations: in our case, we will consider both L 2 (R), represents a high frequency instability and hence manifests itself in the short time instability (local well-posedness) of the underlying solution, while the unstable spectrum near the origin corresponds to instability under long wavelength (low frequency) perturbations, or slow modulations, and hence manifests itself in the long time instability (global well-posedness) of the underlying solution.…”

“…The analysis of stability under arbitrary localized perturbations is more delicate and follows the general modulational theory techniques of Bronski and Johnson [7]: unlike in the solitary wave case, the spectrum of the linearized operator about a periodic wave has a purely continuous L 2 spectrum and hence any spectral instability must come from the essential spectrum. As a result, there are relatively few results in this case.…”

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

“…That is, given (a 0 , E 0 , c 0 ) ∈ D with c 0 > 0, we assume that the Jacobian {T, M, P } a,E,c is non-zero. While these re-parametrization conditions may seem obscure, the non-vanishing of these Jacobians has been seen to be essential in both the spectral and non-linear stability analysis of periodic gKdV waves in [BrJ,J1,BrJK]. In particular, these Jacobians have been computed in [BrJK] for several power-law nonlinearities and, in the cases considered, has been shown to be generically non-zero.…”

Evans functions for periodic waves on infinite cylindrical domains

“…In the long wavelength limit → ∞, the relation between the Evans functions of periodic waves and the limiting pulse profile was discussed qualitatively in [7] and quantitatively in [17]. In the recent work [2], a parity index was constructed for the 1D gKdV equation, in the spirit of [15] for pulses. Recent work on Krein signatures for 1D periodic waves of Hamiltonian PDEs can be found in [9].…”